The conjectural leading term for optimal spherical $\mathbb{L}_2$-discrepancy

Let $d\geq 2$, let $\mathbb{S}^d$ be the unit sphere, and let $\operatorname{D_C}^{\mathbb{L}_2}(\mathbb{S}^d;N)$ denote the optimal spherical-cap $\mathbb{L}_2$-discrepancy among $N$-point configurations. Let $A_d$ be defined by

$$A_d:=\sqrt{\frac{\mathcal{H}_d(\mathbb{B}^d)}{\mathcal{H}_d(\mathbb{S}^d)}\frac{-C_{-1,d}}{[\mathcal{H}_d(\mathbb{S}^d)]^{-1/d}}}.$$

The discrepancy asymptotic conjecture. If the fundamental conjecture for optimal Riesz energy on spheres holds, then

$$\operatorname{D_C}^{\mathbb{L}_2}(\mathbb{S}^d;N)\sim A_dN^{-1/2-1/(2d)}+\cdots\qquad\text{as }N\to\infty.$$

This connects Stolarsky's invariance principle with the second-order term in optimal Riesz $(-1)$-energy. The assertion is conditional on the preceding energy conjecture, while the known bounds establish the same power of $N$ up to constants.